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arXiv · 2505.19168

Effective resistance in planar graphs and continued fractions

Swee Hong Chan·Alex Kontorovich·Igor Pak
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Abstract

For a simple graph $G=(V,E)$ and edge $e\in E$, the effective resistance is defined as a ratio $\frac{\tau(G/e)}{\tau(G)}$, where $\tau(G)$ denotes the number of spanning trees in $G$. We resolve the inverse problem for the effective resistance for planar graphs. Namely, we determine (up to a constant) the smallest size of a simple planar graph with a given effective resistance. The results are motivated and closely related to our previous work arXiv:2411.18782 on Sedl\'a\v{c}ek's inverse problem for the number of spanning trees.

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Swee Hong Chan, Alex Kontorovich, Igor Pak. 2025-05-25. Effective resistance in planar graphs and continued fractions. https://arxiv.org/abs/2505.19168

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